I'm the one who posted the Schreiber pic (), but linking to some toy construction on Baez blog shouldn't run as a "practical example".
Category theory is mostly a language with a few practical concepts. On it's own (in particular, without topological spaces of some sort), category theory has almost no theorems - that's why you almost can't see theorems of it applied. But the language and way of thinking about things is prevalent.
It also unified lot's of concepts.
For example, the fact that
functions with pairs as arguments
[math] f: X \times Y \to Z [/math]
[math] f(x,y) := x\, \sin(y) [/math]
are in one-one correspondence with functions on one argument which have functions as output
[math] g: X \to Z^Y [/math]
[math] g(x) := y \mapsto x\, \sin(y)[/math]
is the same sort of claim as
[math] 5^{(3\cdot 7)} = (5^7)^3 [/math]
and this is the same sort of claim as
[math] ((X\land Y) \to Z) \leftrightarrow (X\to (Y\to Z)) [/math]
>if from propositions X and Y being true, Z follows, then from X follows that given Y is true, Z is also true
>and the other way around
The theory will bring all those together as the same sort of an adjoint in a Carterian closed category
As an application, it's worth noting how some programming languages have stolen the abstractions to make code more slim.
E.g. in haskell, that one is omnipresent
hackage.haskell.org/package/base-4.9.1.0/docs/Data-Functor.html
